Homework Help:
How do I show that the real numbers are not compact?

A trivial, yet difficult question. How would one prove that the real numbers are not compact, only using the definition of being compact? In other words, what happens if we reduce an open cover of R to a finite cover of R?

I let V be a collection of open subset that cover R
Then I make the assumption that that this open cover can be reduced to a finite subcover.
(Clearly this is not possible) I am struggling to see/show what happens to R when I make this assumption. Should I simply find a counterexample, if so, what could it look like?

I have proven several, similar problems, but this one is so general that its tricky.

Compact: every open cover has a finite subcover.
So what is the definition of non-compact? Now show the reals satisfy this definition. If you can negate logical statements then this problem is not at all tricky.